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### Formulas

In the following, is an approximation in the frame where the initial electron and laser collide head-on and the electron is ultra-relativistic.

p, ,p',k
4-momenta of initial electron, laser photon, final electron and emitted photon, respectively. , , , Energies of initial electron, laser photon, final electron and emitted photon, respectively. Laser energy parameter:  .
n
Number of absorbed laser photon. The kinetic relation holds exactly. Here, q is defined as (p is replaced by p' for q') and is called quasimomentum.
x , (0<x<1)
v
v=x/(1-x), x=v/(1+v). ( ) . Maximum v for given n: . Maximum x for given n: . Laser helicity (-1 or +1) , Initial and final electron helicities ( ) Final photon helicity , `Detector helicity' of the final particles. See section 65 of .  is the effective energy of initial electron in the laser field. Final photon angle.  The argument of the Bessel functions in the following expressions: Number of photons per unit time is The terms involving and simultaneously are ignored, i.e., the correlation of polarization between final particles is ignored. The ultra-relativistic approximation has been applied in the terms related to electron helicity ( and/or ). (Note that the electron helicity is a Lorentz invariant quantity only in the ultra-relativistic limit.)

The sum over the final electron and photon helicities gives The functions are defined by  , , , are identical to , , , divided by in Tsai's paper, although the expressions in his paper look much more complicated.

Once x and n are given, the final momentuma are given, in any frame, by  Here, is the azimuthal angle in a head-on frame (therefore its distribution is uniform in [0, ]) and and are given by where is the completely anti-symmetric tensor ( ). These vectors satisfy The vector in eq.(139) is ill-defined when and are colinear in the original frame. In such a case the spatial part of is an arbitrary unit vector perpendicular to .     Next: Usage Up: Compton Process Previous: Compton Process

Toshiaki Tauchi
Thu Dec 3 17:27:26 JST 1998